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AP®︎/College Statistics
Course: ap®︎/college statistics > unit 10.
- Idea behind hypothesis testing
- Examples of null and alternative hypotheses
- Writing null and alternative hypotheses
- P-values and significance tests
- Comparing P-values to different significance levels
- Estimating a P-value from a simulation
- Estimating P-values from simulations
Using P-values to make conclusions
- (Choice A) Fail to reject H 0 A Fail to reject H 0
- (Choice B) Reject H 0 and accept H a B Reject H 0 and accept H a
- (Choice C) Accept H 0 C Accept H 0
- (Choice A) The evidence suggests that these subjects can do better than guessing when identifying the bottled water. A The evidence suggests that these subjects can do better than guessing when identifying the bottled water.
- (Choice B) We don't have enough evidence to say that these subjects can do better than guessing when identifying the bottled water. B We don't have enough evidence to say that these subjects can do better than guessing when identifying the bottled water.
- (Choice C) The evidence suggests that these subjects were simply guessing when identifying the bottled water. C The evidence suggests that these subjects were simply guessing when identifying the bottled water.
- (Choice A) She would have rejected H a . A She would have rejected H a .
- (Choice B) She would have accepted H 0 . B She would have accepted H 0 .
- (Choice C) She would have rejected H 0 and accepted H a . C She would have rejected H 0 and accepted H a .
- (Choice D) She would have reached the same conclusion using either α = 0.05 or α = 0.10 . D She would have reached the same conclusion using either α = 0.05 or α = 0.10 .
- (Choice A) The evidence suggests that these bags are being filled with a mean amount that is different than 7.4 kg . A The evidence suggests that these bags are being filled with a mean amount that is different than 7.4 kg .
- (Choice B) We don't have enough evidence to say that these bags are being filled with a mean amount that is different than 7.4 kg . B We don't have enough evidence to say that these bags are being filled with a mean amount that is different than 7.4 kg .
- (Choice C) The evidence suggests that these bags are being filled with a mean amount of 7.4 kg . C The evidence suggests that these bags are being filled with a mean amount of 7.4 kg .
- (Choice A) They would have rejected H a . A They would have rejected H a .
- (Choice B) They would have accepted H 0 . B They would have accepted H 0 .
- (Choice C) They would have failed to reject H 0 . C They would have failed to reject H 0 .
- (Choice D) They would have reached the same conclusion using either α = 0.05 or α = 0.01 . D They would have reached the same conclusion using either α = 0.05 or α = 0.01 .
Ethics and the significance level α
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What is Hypothesis Testing?
A statistical hypothesis is an assumption about a population parameter . This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses.
Statistical Hypotheses
The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected.
There are two types of statistical hypotheses.
- Null hypothesis . The null hypothesis, denoted by H o , is usually the hypothesis that sample observations result purely from chance.
- Alternative hypothesis . The alternative hypothesis, denoted by H 1 or H a , is the hypothesis that sample observations are influenced by some non-random cause.
For example, suppose we wanted to determine whether a coin was fair and balanced. A null hypothesis might be that half the flips would result in Heads and half, in Tails. The alternative hypothesis might be that the number of Heads and Tails would be very different. Symbolically, these hypotheses would be expressed as
H o : P = 0.5 H a : P ≠ 0.5
Suppose we flipped the coin 50 times, resulting in 40 Heads and 10 Tails. Given this result, we would be inclined to reject the null hypothesis. We would conclude, based on the evidence, that the coin was probably not fair and balanced.
Can We Accept the Null Hypothesis?
Some researchers say that a hypothesis test can have one of two outcomes: you accept the null hypothesis or you reject the null hypothesis. Many statisticians, however, take issue with the notion of "accepting the null hypothesis." Instead, they say: you reject the null hypothesis or you fail to reject the null hypothesis.
Why the distinction between "acceptance" and "failure to reject?" Acceptance implies that the null hypothesis is true. Failure to reject implies that the data are not sufficiently persuasive for us to prefer the alternative hypothesis over the null hypothesis.
Hypothesis Tests
Statisticians follow a formal process to determine whether to reject a null hypothesis, based on sample data. This process, called hypothesis testing , consists of four steps.
- State the hypotheses. This involves stating the null and alternative hypotheses. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false.
- Formulate an analysis plan. The analysis plan describes how to use sample data to evaluate the null hypothesis. The evaluation often focuses around a single test statistic.
- Analyze sample data. Find the value of the test statistic (mean score, proportion, t statistic, z-score, etc.) described in the analysis plan.
- Interpret results. Apply the decision rule described in the analysis plan. If the value of the test statistic is unlikely, based on the null hypothesis, reject the null hypothesis.
Decision Errors
Two types of errors can result from a hypothesis test.
- Type I error . A Type I error occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level . This probability is also called alpha , and is often denoted by α.
- Type II error . A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of committing a Type II error is called Beta , and is often denoted by β. The probability of not committing a Type II error is called the Power of the test.
Decision Rules
The analysis plan for a hypothesis test must include decision rules for rejecting the null hypothesis. In practice, statisticians describe these decision rules in two ways - with reference to a P-value or with reference to a region of acceptance.
- P-value. The strength of evidence in support of a null hypothesis is measured by the P-value . Suppose the test statistic is equal to S . The P-value is the probability of observing a test statistic as extreme as S , assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis.
The set of values outside the region of acceptance is called the region of rejection . If the test statistic falls within the region of rejection, the null hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance.
These approaches are equivalent. Some statistics texts use the P-value approach; others use the region of acceptance approach.
One-Tailed and Two-Tailed Tests
A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution , is called a one-tailed test . For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located on the right side of sampling distribution; that is, a set of numbers greater than 10.
A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a two-tailed test . For example, suppose the null hypothesis states that the mean is equal to 10. The alternative hypothesis would be that the mean is less than 10 or greater than 10. The region of rejection would consist of a range of numbers located on both sides of sampling distribution; that is, the region of rejection would consist partly of numbers that were less than 10 and partly of numbers that were greater than 10.
- AP Statistics Syllabus
AP Statistics PowerPoints
- AP Statistics Additional Notes
- AP Statistics Answers
Chapter 1 Notes Edition 5
Chapter 2 Notes Edition 5
Chapter 3 PowerPoint 2013-2014 , Chapter-3-notes-pdf-2013-2014
Chapter 4 PowerPoint 2013-2014 , Chapter-4-notes-pdf-2013-2014
Chapter 5, ASA Readings
Chapter 5, Section 1
Chapter 5, Section 2
Chapter 5, Section 3
7.1 powerpoint pdf
7.2 powerpoint pdf
7.3 powerpoint pdf
Chapter 9, Section 1
Chapter 9, Section 2
Chapter 9, Section 3
Confidence Intervals: Section 8.1 , Section 8.2 , Section 8.3
Confidence Intervals pdf : Section 8.1 , Section 8.2 , Section 8.3
Hypothesis Testing: Section 10.2 , Section 12.1 , Section 11.1 , Section 10.4
Hypothesis Testing pdf : Section 9.1 Section 9.2 Section 9.3 Day 1 Section 9.3 Day 2 Section 9.1 & 9.2 Error and Power
Inference for 2 proportions: 2 Proportions – CI & Sig Test
Inference for 2 means: 2 Means – CI & Sig Test
Chi-squared testing: Chi-squared – Goodness of fit , Chi-squared – Independence ,
Inference for Regression: Inference for Regression
PDF files :
Inference for 2 proportions: 2 Proportions
Inference for 2 means: 2 Means
Chi-squared testing: Chi-squared – Goodness of fit , Chi-squared – Independence
Inference for Regression: Inference for Regression
Chapter 6, Sections 1 & 2 , Chapter 6, Section 3
Chapter 7, Section 1 , Chapter 7, Section 2
Chapter 8, Section 1 , Chapter 8, Section 2
Chapter 7, Section 1 , Chapter 7, Section 2 ,
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1. A pharmaceutical company claims that 8% or fewer of the patients taking their new statin drug will have a heart attack in a 5-year period. In a government-sponsored study of 2300 patients taking the new drug, 198 have heart attacks in a 5-year period. Is this strong evidence against the company claim?
2. Is Internet usage different in the Middle East and Latin America? In a random sample of 500 adults in the Middle East, 151 claimed to be regular Internet users, while in a random sample of 1000 adults in Latin America, 345 claimed to be regular users. What is the P -value for the appropriate hypothesis test?
3. What is the probability of mistakenly failing to reject a false null hypothesis when a hypothesis test is being conducted at the 5% significance level (α = 0.05)?
4. A research dermatologist believes that cancers of the head and neck will occur most often of the left side, the side next to a window when a person is driving. In a review of 565 cases of head/neck cancers, 305 occurred on the left side. What is the resulting P -value?
5. Suppose you do five independent tests of the form H 0 : µ = 38 versus H a : µ > 38, each at the α = 0.01 significance level. What is the probability of committing a Type I error and incorrectly rejecting a true null hypothesis with at least one of the five tests?
7. Thirty students volunteer to test which of two strategies for taking multiple-choice exams leads to higher average results. Each student flips a coin, and if heads, uses Strategy A on the first exam and then Strategy B on the second, while if tails, uses Strategy B first and then Strategy A. The average of all 30 Strategy A results is then compared to the average of all 30 Strategy B results. What is the conclusion at the 5% significance level if a two-sample hypothesis test, H 0 : µ 1 = µ 2 , H a : µ 1 ≠ µ 2 , results in a P -value of 0.18?
8. Choosing a smaller level of significance, that is, a smaller α-risk, results in
9. The greater the difference between the null hypothesis claim and the true value of the population parameter,
10. A company selling home appliances claims that the accompanying instruction guides are written at a 6th grade reading level. An English teacher believes that the true figure is higher and with the help of an AP Statistics student runs a hypothesis test. The student randomly picks one page from each of 25 of the company's instruction guides, and the teacher subjects the pages to a standard readability test. The reading levels of the 25 pages are given in the following table:
Assuming that the conditions for inference are met, is there statistical evidence to support the English teacher's belief?
11. Suppose H 0 : p = 0.4, and the power of the test for the alternative hypothesis p = 0.35 is 0.75. Which of the following is a valid conclusion?
12. A factory is located close to a city high school. The manager claims that the plant's smokestacks spew forth an average of no more than 350 pounds of pollution per day. As an AP Statistics project, the class plans a one-sided hypothesis test with a critical value of 375 pounds. Suppose the standard deviation in daily pollution poundage is known to be 150 pounds and the true mean is 385 pounds. If the sample size is 100 days, what is the probability that the class will mistakenly fail to reject the factory manager's false claim?
13. For which of the following is a matched pairs t -test not appropriate?
14. Do high school girls apply to more colleges than high school boys? A two-sample t -test of the hypotheses H 0 : µ girls = µ boys versus H a : µ girls > µ boys results in a P -value of 0.02.
Which of the following statements must be true?
I.A 90% confidence interval for the difference in means contains 0.
II.A 95% confidence interval for the difference in means contains 0.
III.A 99% confidence interval for the difference in means contains 0.
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Onward! We use p -values to make conclusions in significance testing. More specifically, we compare the p -value to a significance level α to make conclusions about our hypotheses. If the p -value is lower than the significance level we chose, then we reject the null hypothesis H 0 in favor of the alternative hypothesis H a .
Two-Sample Approach: A response that indicates that the distribution of irritability scores for each of the two samples, placebo AND omega-3, are not badly skewed may satisfy component 2. If the response correctly reports the value of the t-statistic. as 2.256, then the response may satisfy component 3.
AP Statistics: Choosing the Correct Hypothesis Test . Data is Means Test Statistic is t 1 Sample? 2 Samples? 1 Sample t Test (t Test) If Independent Use 2-Sample t Test If paired Find differences, use t-Test . Data is Proportions Test Statistic is z 1 Sample? 1-Prop z Test . 2 Samples? 2-Prop z Test . Categorical Data Test Statistic is χ2 1 ...
AP Statistics: Hypothesis Testing. Objective: To understand the terminology used in hypothesis testing. Warm-up Question1. A candy company claims that its jelly bean mix contains 23% blue jelly beans. Suppose that the candies are packaged at random in small bags containing about 350 jelly beans. Find the mean and standard deviation
Courses on Khan Academy are always 100% free. Start practicing—and saving your progress—now: https://www.khanacademy.org/math/ap-statistics/chi-square-tests/...
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AP Statistics - Chapter 9 Notes: Testing a Claim. 9.1: Significance Test Basics. Null and Alternate Hypotheses. The statement that is being tested is called the null hypothesis (H0). The significance test is designed to assess the strength of the evidence against the null hypothesis. Usually the null hypothesis is a statement of "no effect ...
The AP Statistics curriculum is designed primarily to help students understand statistical concepts and become critical consumers of information. Being able to perform statistical computations is of, at most, secondary importance and for some topics, such as power, is not expected of students at all. ... Of the hypothesis tests in the AP ...
It is designed to help you master the Advanced Placement Statistics Exam. > Begin lesson 1. About the Tutorial. This tutorial provides accurate and complete coverage of the AP Statistics curriculum. Specifically, the AP Statistics curriculum and this tutorial cover the following topics: Exploring data.
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There are 5 main steps in hypothesis testing: State your research hypothesis as a null hypothesis and alternate hypothesis (H o) and (H a or H 1 ). Collect data in a way designed to test the hypothesis. Perform an appropriate statistical test. Decide whether to reject or fail to reject your null hypothesis. Present the findings in your results ...
A hypothesis test is used to see whether an assumption is statistically plausible by using sample data The basic formula for a hypothesis test is: Statistic - Parameter/ Standard Deviation of Statistic The higher the Z or t score, the lower the p value, and the more evidence there is to reject the null hypothesis Five Steps 1. Hypothesis 2.
This process, called hypothesis testing, consists of four steps. State the hypotheses. This involves stating the null and alternative hypotheses. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false. Formulate an analysis plan.
Therefore it is reasonable to assume that the sample sizes are less than 10 percent of the respective population sizes. Question 4 (continued) Using the pooled estimate of the proportion of resistant plants, p ˆ c ≈ 0.2835, the value of the test statistic is: = z 0.385 − 0.197 ≈ 2.21 (0.2835)(0.7165) 61.
AP Statistics PowerPoints. Unit 1: Chapter 1 Notes Edition 5. Chapter 2 Notes Edition 5. Unit 2: Chapter 3 PowerPoint 2013-2014, ... Hypothesis Testing: Section 10.2, Section 12.1, Section 11.1, Section 10.4. Hypothesis Testing pdf: ...
6 AP Statistics Formula Sheet 8 Part 1: Exploring Data 8 Variable Types 85‐Number Summary 8Other Terms to Know 9 Frequency Distribution ... 54 Hypothesis Testing 55 Pooled t‐Interval and t‐Test 56 Chapter 25: Inferences About Paired Data 56 Pairing 56 Sample of Means
a test that determines whether to accept or reject a hypothesis based upon the given results. Hypothesis test. Tells if a sample statistic (p-hat) occurs by random occurance or not (aka statisticlaly significant) Assumptions, hypothesis statements and parameters, calculations, conclusion. Step for a hypothesis test.
The response does not satisfy component 3 because, although the response correctly names the test, the z-statistic formula provided is not correct. Section 1 was scored partially correct (P). Question 4 (continued) In section 2 the response satisfies components 1 and 2 by correctly checking the conditions.
An English teacher believes that the true figure is higher and with the help of an AP Statistics student runs a hypothesis test. The student randomly picks one page from each of 25 of the company's instruction guides, and the teacher subjects the pages to a standard readability test. The reading levels of the 25 pages are given in the following ...